Non-Differentiable Embedding of Lagrangian structures

Transcription

1 Non-Differentiable Embedding of Lagrangian structures Isabelle Greff Joint work with J. Cresson Université de Pau et des Pays de l Adour CNAM, Paris, April, 22nd 2010

2 Position of the problem 1. Example We consider particles moving continuously along a path x(t), of mass 1, under a potential field U. The trajectory is given by the Newton equation d 2 x(t) = U(x). dt2 2 Assumption: the trajectories are smooth. Example 3 Work in Physics from L. Nottale: no hypothesis concerning the differentiability. Natural trajectories are everywhere non-differentiable. Nottale s idea: take into account this loss of differentiability on the micro-scale. 4 Idea Extension of the notion of derivative.

3 5 Non-differentiable embedding an ODE is the restriction of a more general differentiable equation. Non-differentiable embedding of ODE. Conservation of the structure of the original ODE by the embedding procedure? 6 Newton s equation derives from a variational principle associated to a function L called Lagrangian and given here by L(t, x, v) = 1 2 v2 U(x). In fact the trajectories solution of the Newton equation are extremals of the Lagrangian functional L defined by L(x) = b a L(t, x(t), x (t))dt. Lagrangian system is the input of a Lagrangian and a variational principle also called least-action principle.

4 Outline Diagram 1. Classical calculus of variations 2. Non-differentiable embedding Quantum calculus Non-differentiable embedding of Lagrangian systems Non-differentiable calculus of variations Coherence principle 3. Application to Navier-Stokes equation 4. Noether s theorem

5 Notations Let d N, I be an open set in R, and a, b R, a < b, such that [a, b] I. Let F(I, R d ) the set of functions defined in I taking value in R d. Let C 0 (I, R d ) (respectively C 0 (I, C d )) be the set of continuous functions x : I R d (respectively x : I C d ). Let n N, and C n (I, R d ) (respectively C n (I, C d )) be the set of functions in C 0 (I, R d ) (respectively C 0 (I, C d )) which are differentiable up to order n.

6 Hölderian functions Let w C 0 (I, R d ). Let t I. 1. w is Hölder of Hölder exponent α, 0 < α < 1, at point t if c > 0, η > 0 s.t. t I t t η w(t) w(t ) c t t α, where is a norm on R d. 2. w is α-hölder and inverse Hölder with 0 < α < 1, at point t if c, C R +, c < C, η > 0 s.t. t I t t η c t t α w(t) w(t ) C t t α Example H α (I, R d ) := {x C 0 (I, R d ), x is α Hölder and inverse Hölder}. Let C k α (I, C d ) C 0 (I, C d ) defined by: Example C k α (I, C d ) := {x : C 0 (I, C d ), x(t) := u(t) + w(t), u C k (I, C d ), w H α (I, C d )}.

7 Calculus of variations (1) We consider admissible Lagrangian functions L L : R R d R d C (t, x, v) L(t, x, v) such that L(t, x, v) is holomorphic with respect to v, differentiable with respect to x. Example L(t, x, v) = 1 2 v2 U(x), with U is a function of x. A Lagrangian function defines a functional on C 1 (I, R), denoted by L : C 1 (I, R d ) R x L(x) := b Let V be the space of variations defined by: a L ( s, x(s), dx dt (s)) ds. V := {h C 1 (I, R d ), h(a) = h(b) = 0}.

8 Calculus of variations (2) A functional L is differentiable at point γ C 2 (I, R d ) if and only if L(γ + θh) L(γ) = θdl(γ)(h) + o(θ), for θ > 0 sufficiently small and any h V. DL(γ)(h) is the Gâteaux derivative of L at point γ in the direction h. An extremal for the functional L is a function γ C 2 (I, R d ) such that DL(γ)(h) = 0 for any h V. Theorem The extremals of L coincide with the solutions of the Euler-Lagrange equation denoted by (EL) and defined by d dt [ L v ( t, γ(t), dγ dt (t) )] = L x ( t, γ(t), dγ ) dt (t). (EL)

9 Quantum derivative (1) Idea: Extension of the classical notion of derivative. Let x C 0 (I, R d ). For all ɛ > 0, we call ɛ-left and right quantum derivatives the quantities d + ɛ x(t) := x(t + ɛ) x(t), d ɛ x(t) := ɛ x(t) x(t ɛ), ɛ Definition For any ɛ > 0, the ɛ-scale derivative of x at point t is the quantity defined for µ {1, 1, 0, i, i} by ɛ : C0 (I, R d ) C 0 (I, C d ) x ɛx where ɛx (t) := 1 [ (d + 2 ɛ x(t) + d ɛ x(t) ) + iµ ( d + ɛ x(t) d ɛ x(t) )] t I.

10 Remarks If x C 1 (I, R d ɛ x ), then lim ɛ 0 = dx the classical derivative of x. dt For µ = i, ɛ = d ɛ For µ = i, ɛ = d+ ɛ Allows to recover the backward and forward derivatives. Extension for x C 0 (I, C d ) by ɛ x (t) := ɛre(x) + i ɛim(x), (1) where Re(x) and Im(x) are the real and imaginary part of x. composition of ɛ

11 Quantum derivative 2 Idea: Build an analogous of the derivative for non-differentiable functions. Construction We consider C 0 (I ]0, 1], R d ) the space of continuous functions f : I ]0, 1] R d (t, ɛ) f(t, ɛ) Let C 0 conv(i ]0, 1], R d ) be a subspace of C 0 (I ]0, 1], R d ): C 0 conv(i ]0, 1], R d ) := {f C 0 (I ]0, 1], R d ), lim f(t, ɛ) exists for any t I}. ɛ 0 Let E be a complementary space of C 0 conv(i ]0, 1], R d ) in C 0 (I ]0, 1], R d ). Let π be the projection onto C 0 conv(i ]0, 1], R d ) defined by π : C 0 conv(i ]0, 1], R d ) E C 0 conv(i ]0, 1], R d ) f conv + f E f conv.

12 Quantum derivative 3 We can then define the operator. by. : C 0 (I ]0, 1], R d ) F(I, R d ) f π(f) : t lim ɛ 0 π(f)(t, ɛ). Definition Let us introduce the new operator (without ɛ) on the space C 0 (I, R d ) by: x := π( ɛx ) (2) For x C 1 (I, R d ), then x(t) = dx dt (t). For w H α (I, R d ), then w(t) = 0. (Since c.ɛ α 1 ɛ w(t) ) For x C 1 α (I, C d ), 0 < α < 1, with x := u + w, then x(t) = u (t).

13 Properties Quantum derivative 4 Non-differentiable Leibniz rule Let f be α-hölder and g be β- Hölder, with α + β > 1, f (f g) = g + f g. Composition Let f be a C 2 (R d I, R) function. Let 1 2 α < 1. Let x = (x 1,..., x d ) C 1 α (I, R d ) written as x := u + w where u = (u 1,..., u d ) C 1 (I, R d ) and w = (w 1,..., w d ) H α (I, R d ), then the following formula holds a k,l (x(t)) := π f(x(t), t) ɛ 2 = x f(x(t), t) u(t) + f (x(t), t) t d d 1 2 f + (x(t), t) a k,l (w(t)), 2 x k x l k=1 l=1 + `(d ɛ x k (t))(d + ɛ x l (t))(1 + iµ) (d ɛ x k (t))(d ɛ x l (t))(1 iµ).

14 Non-differentiable embedding of operators 1. We denote by O the differential operator acting on C n (I, C d ) defined by n O = F i ( d i dt i G ) i, (3) i=0 where is the standard product of operators and the usual composition, i.e. (A B)(x) = A(B(x)), with the convention that ( ) 0 d = Id, where Id denotes the identity mapping on C. dt 2. The non-differentiable embedding of O written as (3), denoted by Emb (O) is the operator Emb (O) = n i=0 F i ( i i G ) i. (4) Remark: In the rest of the talk we will consider curves x C 0 (I, R d ), such that x C0 (I, R d ) or smooth enough.

15 Non-differentiable embedding of ODE 1. Let the ordinary differential equation associated to O be defined by O ( x, dx dt,..., dk x) = 0, for any x C k+n dt k (I, C). (5) 2. The non-differentiable embedding of equation (5) is defined by Emb (O) (x, x (,..., k x i ) x ) = 0, x, k i C 0 (I, C d ). 1 i k (6)

16 The non-differentiable embedded Euler-Lagrange equation 1. The Euler-Lagrange equation (EL) is: [ ( d L t, γ(t), dγ )] dt v dt (t) = L ( t, γ(t), dγ ) x dt (t). (EL) 2. Let O (EL) be the associated non-differentiable embedded operator O (EL) := d dt L v L x The Euler-Lagrange equation (EL) is O (EL) ((t, γ(t), dγ dt (t)) = The non-differentiable Euler-Lagrange associated to (EL) is then: ( ) L ( t, γ(t), v γ(t)) L ( t, γ(t), x γ(t)) = 0. Emb (EL)

17 Embedding of the Lagrangian functional The Lagrangian functional associated to L is: L : C 1 (I, R d ) R, x C 1 (I, R d ) b a L ( s, x(s), dx dt (s)) ds. The natural embedding of the Lagrangian functional L is given by L : C 0 (I, R d ) R, x C 0 (I, R d ) b a L ( s, x(s), x(s) ) ds, always with x C0 (I, R d ).

18 Non-differentiable calculus of variations Let α, β be real numbers 0 < α, β < 1, s.t. α + β > 1. Let V := {h C 0 (I, R d ), β-hölder, h(a) = h(b) = 0}, be the space of non-differentiable variations. Definition Let Φ : C 0 (I, R d ) C be a functional. The functional Φ is called V -differentiable on a curve γ C 0 (I, R d ), α-hölder if and only if its Gâteaux differential Φ(γ + ɛh) Φ(γ) lim ɛ 0 ɛ exists in any direction h V. And then DΦ is called its differential and is given by Φ(γ + ɛh) Φ(γ) DΦ(γ)(h) = lim. ɛ 0 ɛ

19 V -extremal curves A V -extremal curve of the functional Φ on the space V of curves is a curve γ α-h ölder satisfying DΦ(γ)(h) = 0, for any h V. Theorem The differential of L on γ C 0 (I, R d ), α-hölder and γ α-hölder is given for any h V by DL (γ)(h) = b a ( L ( t, γ(t), γ(t) x ) h(t) + L v ( t, γ(t), γ(t) ) ) h(t) dt.

20 Theorem (Non-differentiable least-action principle) Let 0 < α < 1, α + β > 1 and β α. Let L be an admissible Lagrangian function of class C 2. We assume that γ C 0 (I, R d ) α-hölder and γ α-hölder. The curve γ is an extremal curve of the functional L on the space of variations V, if and only if it satisfies the following generalized Euler-Lagrange equation ( ) L( γ(t) ) L γ(t) t, γ(t), (t, γ(t), ) = 0. (NDEL) x v

21 Coherence Embedding of the Euler-Lagrange equation denoted by Emb(EL). Embedding of the Lagrangian functional L. Non-differentiable calculus of variation leads to a non-differentiable Euler-Lagrange equation N.D EL. Conclusion N.D EL = Emb(EL). We preserve the Lagrangian structure passing to the non-differentiable embedding.

22 Application to the Navier-Stokes equation Extension of the definition of characteristics The classical method of characteristics for a PDE is to look for t x(t) satisfying the following ordinary differential equation d (u(x(t), t)) = F (x(t), t), dt where F is the non homogeneous part of the PDE. Using the operator one can generalize this method. We say that a curve t x(t) is a non-differentiable characteristic for a given PDE if the solution u(x(t), t) satisfies (u(x(t), t) = F (x(t), t), and x and t satisfy an ordinary differential equation in.

23 Let us consider the Navier-Stokes equation u d t + u u k = ν x u x p, x k k=1 where the unknown are the velocity u(t, x) R d, u = (u 1,..., u d ), and the pressure p(t, x) R. The constant ν R + is the viscosity. Theorem The non-differentiable characteristics x Cnav 1 α, 1 2 α < 1 of the Navier-Stokes equations correspond to Cnav 1 α extremals of the Lagrangian L(t, x, v) = 1 2 v2 p(x, t), C 1 α nav := {x = (x 1,..., x d ) C 1 α (I, R d ), x i (t) = t 0 u i (x(s), s) ds + W i (t), W i H α, 1 α < 1, i = 1,..., d}, 2 where u is a solution of the Navier-Stokes equation and W satisfies a l,l (W (t)) = 2ν and a k,l (W (t)) = 0 if k l.

24 Idea of the proof For x Cnav 1 α (I, R d ) we have x (t) = u(x(t), t), and for any i = 1,..., d u i (x(t), t) = x u i (x(t), t) u i (x(t), t) + u i (x(t), t) + t d d 1 2 u i (x(t), t)a k,l (w(t)). 2 x k x l k=1 l=1 The non-differentiable caracteristics are curve t x(t) such that (u(x(t), t)) = xp. this equation can be rewritten as ( ) x = x p. which is the non-differentiable Euler-Lagrange equation associated to L.

25 Noether s theorem (1) We call {φ s } s R a one parameter group of diffeomorphisms φ s : R d R d, of class C 1 satisfying φ 0 = Id, φ s φ u = φ s+u, φ s is of class C 1 with respect to s. Invariance Let Φ = {φ s } s R a one parameter group of diffeomorphisms. An admissible Lagrangian L is said to be invariant under the action of Φ if ( L t, x(t), dx dt (t) ) ( = L t, φ s (x(t)), d dt( φs (x(t)) )), s R, t R, for any solution x of the Euler-Lagrange equation.

26 Noether s theorem (2) First Integral A first integral for the Euler-Lagrange equation is a function J : R R d R d R such that for any solution x of the Euler-Lagrange equation, d ( ) J(t, x(t), ẋ(t)) = 0 for any t R. dt Noether s theorem Let L be an admissible Lagrangian of class C 2 invariant under Φ = {φ s } s R, a one parameter group of diffeomorphisms. Then, the function J : (t, x, v) L v (t, x, v) dφ s(x) s=0 ds is a first integral of the Euler-Lagrange equation (EL).

27 Passage to the non-differentiable case? invariance of Lagrangian N.D. Embinvariance of N.D Lagrangian Noether s thm. First integral

28 Passage to the non-differentiable case? invariance of Lagrangian Noether s thm. First integral N.D. Emb invariance of N.D Lagrangian N.D. Emb N.D. First integral

29 Passage to the non-differentiable case? N.D. Emb invariance of Lagrangian invariance of N.D Lagrangian Noether s thm. N.D. Noether s thm. First integral N.D. Emb N.D. First integral

30 Non-differentiable Noether s theorem (1) -invariance Let Φ = {φ s } s R be a one parameter group of diffeomorphisms. An admissible Lagrangian L is said to be -invariant under the action of Φ if L(t, x(t), x (t)) = L(t, φ s(x(t)), (φ s(x(t)))), s R, t I. for any solution x C 1 of the non-differentiable Euler-Lagrange equation (NDEL). Persistence of invariance? Sufficient condition: Let Φ = {φ s } s R be a one parameter group of diffeomorphisms, such that φ s : C d C d satisfies the -commutation property: ( x ) (φ s(x)) = φ s, s R. (7) If L is strongly invariant i.e: L(t, x, v) = L(t, φ s (x), φ s (v)), s R, t I, x R d, v R d. Then, L is -invariant under the action of Φ = {φ s } s R.

31 Non-differentiable Noether s theorem (2) Let φ be a linear map, then φ satisfies the property of -commutation. A generalized first integral associated to the non-differentiable Euler-Lagrange equation is a function J : R R d C d C such that for any solution x of (NDEL), we have ( J ( t, x(t), x(t) ) ) = 0 t R. Non-differentiable Noether s theorem Let L be a Lagrangian of class C 2 -invariant under Φ = {φ s } s R, a one parameter group of diffeomorphisms, such that φ s : C d C d, for any s R. Then, the function J : (t, x, v) L v (t, x, v) dφ s(x) s=0 (8) ds is a generalized first integral of the non-differentiable Euler-Lagrange equation (NDEL) on H α (I, R d ) with 1 2 < α < 1.

32 Conclusion The non-differential embedding preserves the Lagrangian structure Solutions of the Navier-Stokes seen as extremals of a non-differentiable Lagrangian Coherence for the Hamiltonian systems Persistence of the invariance of the Lagrangian under special conditions. Non-differentiable Noether theorem

33 Example of inverse-hölder function: Tagaki-Knopp Retour

34 Example of smooth curve Retour

35 Example of non-smooth curve Retour

36 Outline Diagram Lagrangian functional L N.D. EmbN.D. Lagrangian functional least-action principle Retour Newton s equation

37 Outline Diagram Lagrangian functional L least-action principle Newton s equation N.D. Emb N.D. Newton s equation Retour

38 Outline Diagram Lagrangian functional L least-action principle Newton s equation Retour N.D. Emb N.D. Lagrangian functional L N.D. Emb N.D. Newton s equation

39 Outline Diagram N.D. Emb Lagrangian functional L N.D Lagrangian functional L least-action principle Newton s equation Retour N.D. Emb N.D. least-action principle N.D. Newton s equation